The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $15.7$ years; the standard deviation is $2.2$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living less than $22.3$ years.
Solution: $15.7$ $13.5$ $17.9$ $11.3$ $20.1$ $9.1$ $22.3$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $15.7$ years. We know the standard deviation is $2.2$ years, so one standard deviation below the mean is $13.5$ years and one standard deviation above the mean is $17.9$ years. Two standard deviations below the mean is $11.3$ years and two standard deviations above the mean is $20.1$ years. Three standard deviations below the mean is $9.1$ years and three standard deviations above the mean is $22.3$ years. We are interested in the probability of a sloth living less than $22.3$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the sloths will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the sloths will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $9.1$ years and the other half $({0.15\%})$ will live longer than $22.3$ years. The probability of a particular sloth living less than $22.3$ years is ${99.7\%} + {0.15\%}$, or $99.85\%$.